Combinatorics Seminar: "Filtrations and bases for the cohomology of regular nilpotent Hessenberg varieties"

Abstract: Hessenberg varieties are subvarieties of the flag variety with connections to representation theory, algebraic geometry, and combinatorics.  In 2015, Brosnan and Chow proved the Shareshian--Wachs conjecture, linking the Stanley--Stembridge conjecture in combinatorics to the geometry of Hessenberg varieties through Tymoczko's symmetric group action on the cohomology of regular semisimple Hessenberg varieties.  The $S_n$-invariant subspace of this representation is isomorphic to the cohomology of a regular nilpotent Hessenberg variety.  This talk will give a brief overview of that story and present a new filtration for the cohomology rings of regular nilpotent Hessenberg varieties.  As an application, we obtain a monomial basis for this ring.  This is joint work with M. Harada, T. Horiguchi, S. Murai, and J. Tymoczko.